We follow [1]}. An embedding \(i:G \rightarrow S\)  of a graph \(G\)  into a connected, compact surface \(S\)  is a 1-1 continuous map. Two embeddings \(i_1\)  and \(i_2\)  of \(G\)  in a surface \(S\)  are equivalent if there exists a homeomorphism \(h: S\rightarrow S \)  such that \(h \circ i_1 = i_2\) . A 2-cell embedding \(i: G \rightarrow S\)  is an embedding for which the components of the set \(S\setminus i(G)\)  are homeomorphic to open discs, the 2-cells.
A 2-cell embedding \(i:G \rightarrow S\)  induces a (cell) decomposition \(K=(V,E,F)\)  of \(S\) , where \(i(G)=(V,E)\)  in the obvious way and \(F\)  are the 2-cells of \(S\setminus i(G)\) . Given a 2-cell \(\Sigma \) , the edges and vertices in the boundary of \(\Sigma \)  can be arranged as a closed walk \(\sigma _1,\sigma _2,\ldots ,\sigma _k\)  along the edges of \(G\) , the boundary walk of \(\Sigma \) .
